# Cubes and Cube Roots

Cube and Cube Roots are fundamental concepts in algebra and it is introduced in the early classes. The multiplication of a number to itself gave rise to a square and then if we multiply the number by its square then the result becomes a cube and the inverse of the cube is the cube root which we will study in this article. In this article, we will learn about cubes and cubes and also learn about the methods to find both cubes of a number and cube roots of a number. So, let’s dive into the concept of Cubes and Cube Roots.

## What is Cube?

Physically a cube is a solid figure which has all its sides equal. In mathematical terms, the cube of a number is the result of multiplying a whole number by itself twice, i.e., the whole number is used three times just like the sides of a cube. For example:

### Cube 1 to 20

The cube of natural numbers 1 to 20 is discussed in the table below,

## Cube of Negative Numbers

Similar to positive integers we can find the cube of negative integers as well. It is the same for negative numbers as well. To find the cube of a negative number we just multiply the negative number three times by itself. As a product of two negative numbers is positive and the product of negative and positive numbers is negative. Thus, a cube of a negative number is always negative.

For example, a cube of -2 is nothing but (-2)×(-2)×(-2) = 4×(-2) = -8.

## Cube of Fractions

As we discussed in the properties of the cube

## Perfect Cubes

A perfect cube is a positive integer equal to some other positive integer raised to the third power. In other words, when we multiply an integer by itself the resulting number is called a perfect cube. For example, 125 is a perfect cube since 125 = 5 × 5 × 5 = 53. However, 121 is not a perfect cube because there is no integer n such that n3 = 121 = 112. Some other examples of perfect cubes are 1, 8, 27, 64, 125, 216, 343, . . . , etc.

## What is Cube Root?

The cube root of a number is a value that, when multiplied three times, gives that number. For example, 3 × 3 × 3 = 27, so the cube root of 27 is 3.

### Symbol of Cube Root

The cube root formula is the formula that is used to find the cube root of any substance. Suppose the cube of a number is x is y we can represent this as,

x3 = y

Now the cube root of y is calculated as,

y1/3 = x

The image added below represent the same.

## How to find the Cube Root of a Number?

Now, let’s learn about them in detail.

### By Prime Factorization

Prime factorization is a method through which you can easily determine whether a particular figure represents a perfect cube. If each prime factor can be clubbed together in groups of three, then the number is a perfect cube.

For example: Let us consider the number 1728.

As prime factorization of 1728 is 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, which can be rewritten as 23 × 23 × 33.

Here, all the numbers can be clubbed into groups of three. So we can definitely say that 1728 is a perfect cube. In fact, the cube root of 1728 is a product of numbers taken one from each group i.e.,  2 × 2 × 3 = 12.

Example 1: Find the cube root of 216.

216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 * 33 = 63

Hence, the cube root of 216 is 6.

Example 2: Find the cube root of 5832.

5832 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

5832 = 23 × 33 × 33 = 183

Hence, the cube root of 5832 is 18.

### Cube Root of a Cube Number using Estimation

Step 1: Take any cube number say 117649 and start making groups of three starting from the rightmost digit of the number.

So 117649 has two groups, and the first group is (649) and the second group is (117).

Step 2: The unit’s digit of the first group (649) will decide the unit digit of the cube root. Since 649 ends with 9, the cube root’s unit digit is 9.

Step 3: Find the cube of numbers between which the second group lies. The other group is 117.

We know that 403= 64000 i.e., second group for cube of 40 is 64, and 503= 125000 i.e., second group for 50 is 125. As 64 < 117 < 125. Thus, the ten’s digit of the requred number is either 4 or 5 and 50 is the least number with 5 as ten’s digit. Thus, 4 is the ten’s digit of the given number.

So, 49 is the cube root of 117649.

Example: Estimate the Cube root of the number 357911.

Let’s take another cube number, say 175616.

Step 1: Starting from the rightmost digit, group the digits in threes. So, the first group is (616) and the second group is (175).

Step 2: The unit digit of the first group (616) is 6, which corresponds to the unit digit of the cube root 6.

Step 3: Find the cubes of numbers between which the second group lies. We know that 43³ = 79507 and 44³ = 85184. Since 175 is between 79507 and 85184, the tens digit of the required number is 4.

Therefore, the cube root of 175616 is 46.

### Hardy-Ramanujan Numbers

Numbers like 1729, 4104,13832, etc. are known as Hardy – Ramanujan Numbers because they can be expressed as the sum of two cubes in two different ways. A number n is said to be Hardy-Ramanujan Number if

n = a3 + b3 = c3 + d3

The first four Hardy-Ramanujan numbers are:

91 = 63 + (-5)3 = 43 + 33

Note: The number 1729 is also sometimes referred to as the taxicab number as it was the number of the taxi taken by Dr. Hardy while going to meet Ramanujan at the hospital.

## FAQs on Cube and Cube Roots

3D Cube is solid and a physical object as cube numbers are the abstract concept which is a number we get after raising any number to it’s third power.

Cube of 3 is 27.

Cube of 4 is 64.

### How do you calculate the cube root of a number?

We can find the cube root of a perfect cube number using the prime factorization method or we can estimate the cube root using the steps explained in this same article.

### What is the Cube Root of 27?

A cube of 3 is 27, therefore cube root of 27 is 3.

### What is the Cube Root of 125?

A cube of 5 is 125, therefore cube root of 125 is 5.

### What is the relationship between Cubes and Cube roots?

Cube and Cube Roots are inverse operations that undo the effect of each other on any number. In other words, if we take any number and take it’s third power to find its cube and then find the cube root of the result. We will end up with the same number we have taken at first.

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Given a number n, find the cube root of n.Examples:

Input: n = 3
Output: Cubic Root is 1.442250

Input: n = 8
Output: Cubic Root is 2.000000

We can use binary search. First we define error e. Let us say 0.0000001 in our case. The main steps of our algorithm for calculating the cubic root of a number n are:

Below is the implementation of above idea.

## C++

using namespace std;

double diff(double n,double mid)

return ((mid*mid*mid) — n);

double cubicRoot(double n)

double start = 0, end = n;

double e = 0.0000001;

double mid = (start + end)/2;

double error = diff(n, mid);

if (error <= e)

end = mid;

start = mid;

double n = 3;

printf(«Cubic root of %lf is %lf
«,

## Java

static double diff(double n,double mid)

static double cubicRoot(double n)

System.out.println(«Cube root of «+n+» is «+cubicRoot(n));

## Python3

def diff(n, mid) :

return (n — (mid * mid * mid))

return ((mid * mid * mid) — n)

def cubicRoot(n) :

while (True) :

mid = (start + end) / 2

if (error <= e) :

print(«Cubic root of», n, «is»,

static double diff(double n, double mid)

return (n-(mid * mid * mid));

return ((mid * mid * mid) — n);

double mid = (start + end) / 2;

public static void Main ()

Console.Write(«Cube root of «+ n

+ » is «+cubicRoot(n));

## PHP

return (\$n — (\$mid *

\$mid * \$mid));

return ((\$mid * \$mid *

\$mid) — \$n);

\$start = 0;

\$end = \$n;

\$e = 0.0000001;

\$mid = ((\$start + \$end)/2);

\$error = diff(\$n, \$mid);

if (\$error <= \$e)

\$end = \$mid;

\$start = \$mid;

\$n = 3;

## Javascript

function diff(n, mid)

let start = 0, end = n;

let e = 0.0000001;

let mid = (start + end)/2;

let error = diff(n, mid);

let n = 3;

document.write(«Cube root of «+n+» is «+cubicRoot(n));

Cubic root of 3.000000 is 1.442250

Time Complexity: O(logn)

The cube root of a number is an important concept like square roots in the number system. Note that the cube root is the inverse method of finding cubes. In this section, we will discuss about the cube root of a number.

What is cube and cube root? The number obtained by multiplying a given number two times by itself is called the of that given number. So is the cube of . Here the number is called the cube root of .

#### Definition of Cube of a Number

A number is called the cube of a number if the number is a product of three numbers of ‘s. Thus is the cube of .

For example, is the cube of as .

Similarly, is the cube of as .

#### Definition of Cube Root

Thus, if then we say is the cube root of .

Note that , so we can say that is a cube root of .

#### Cube Root Symbol

More examples of cube roots:

#### Rules of Cube Root

Perfect cube table:

#### Simplifying Cube Roots

Let us now learn how to simplify cube roots. We will simplify cube root of .

At first, we will factorize . See that

24 = 2×2×2×3

Taking cube root on both sides, we get

#### Methods of Finding Cube Root

How to find the cube root of a number?

#### Addition or Subtraction of Cube Roots

First, we express the cube roots into their simplified forms as above like cube root of .

Firstly, we will simplify both cube roots.

#### Multiplication or Division of Cube Roots

We write the cube roots into their simplified forms.

#### Cube Root of a Decimal

At first, we need to express the decimal number as a fraction.

Now we will simplify the cube roots of \$x\$ and \$y\$ and then we put their values in the fraction obtained in step 2.

Simplifying the fraction obtained in step 3, we will get the cube root of the given decimal number.

To understand the above method, we provide an example here.

Taking square root we get

So the square root of is .

#### Cube Root of a Complex Number

The computation of the cube of a complex number is not as simple as the method of finding the cube root of a real number. We know that the general form of a complex number is \$a+ib,\$ where both \$a\$ and \$b\$ are real numbers.

#### Cube Root as a Function

The above function \$f\$ is one-to-one but not onto.

#### Generalization of Cube Roots

Let \$f(x)\$ be a polynomial with a root \$c\$. Then we must have \$f(c)=0.\$ This type of roots are classified as polynomial roots.

The computation of cube roots has many applications in several branches of mathematics; such as

#### Solved Problems of Cube Roots

Find the cube root of \$125.\$

Note that \$125=5 imes 5 imes 5\$

So the value of the cube root of is .

#### Cube Root of Numbers

Derivative of. In this article, we will find the derivative of the cube root of by applying the power rule of derivatives. We will also use the limit definition to evaluate the derivative of the cube root of .

## What is the derivative of cube root of x?

First Method of Finding Derivative of Cube Root of x: At first, we will calculate the derivative of cube root x by the power rule of derivatives. See the below steps.

Therefore, we get that

Finally, simplify the above expression. Thus from we have

Differentiating both sides with respect to \$x\$, we obtain that

Also Read: Derivative of Square Root of x

Now, we will find the derivative of cube root of \$x\$ by the first principle.

## Derivative of cube root of x from first principle

Myltiplying the numerator and the denominator of by the above quantity, we get that

Thus, the derivative of the cube root of by first principle is

## Derivative of Cube Root x by Logarithmic Differentiation

Now, we will find the derivative of the cube root of x with the help of the logarithmic derivative. Note that cube root x can be written as . Write

We will take natural logarithms on both sides (i.e, logarithm with base e, denoted by ln). Thus we get that

ln y = 1/3 ln x

Differentiating with respect to x, we have

## Application of Derivative of cube root of x

From above we get the derivative of root \$x\$ which is given below.

This will help us to find the derivatives of many functions involving cube roots by the chain rule of derivatives. At first, we will calculate the derivative of the cube root of .

Next, we will find the derivative of the cube root of , that is, the derivative of

Q1: What is the derivative of cube root of x?

Ans: The derivative of cube root of x is 1/3×2/3.

Q2: How a cube root of x defined?

The cube root of is a number when multiplied by itself two times will be . If a cube has a volume of 125 unit3, then we use the value of the cube root of to find the length of the cube.

Let us now calculate the cube root of

We know that 125=5×5×5. As this is a product of three number of ‘s. we can write

Taking cube root on both sides, we have

As cube root can be written as power , we get that

= 53×1/3   as we know that (am)n = am×n

= 51 = 5

So the cube root of is .

#### Is 125 a perfect cube number?

Note that the value of the cube root of is . To prove this by the prime factorization method, we will at first factorize

As has the unit digit , it will be divisible by . So we have

By the above logic, we write

So finally we get

Note that the above is the prime factorization of .

Taking cube root on both sides of , we get that

∴  the cube root of is .

#### Is Cube Root of 125 Rational?

By definition, we know that a rational number can be expressed as where and q are integers with .

So 5 can be written as p/q with .

is a rational number.

⇒ the cube root of is also a rational number.

Conclusion: The cube root of 125 is not an irrational number.

Q1: Find the cube root of 125.

Answer: As 125 = 5×5×5, the cube root of 125 is 5.

Q2: Is 125 a perfect cube number?

Answer: Yes, 125 is a perfect cube number as 125=53.

To know the cube root formula, you need to know the cube formula.

Cube of any digit, forms by multiplying the digit by itself three times. For instance, to find the cube:

We need to multiply 5 three times in the case of : 5 × 5 × 5 = 125

Note: We need to write down “5 cube” as (the little 3 on the top means the number appears three times during the multiplication process.)

The Cube Formula for any value ‘x’ can be given as,

### What are Perfect Cubes?

If we group the prime factors of a number in triples of equal factors, then that number is known as a perfect cube. In order to check whether a number is a perfect cube or not, we need to find its prime factors and then group together triplets of the prime factors. If no factor is left out after the process then the number is known to be a perfect cube.

Here are a few more examples of perfect cube numbers:

### What is a Cube Root?

A cube root goes the opposite direction. For example, 3 is cubed to give the result 27 so the cube root of 27 will be 3.

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Therefore, the cube root of a number is a special number which when cubed gives the original number as a result. The cube root of 27 is 3 because 3 is cubed to produce 27.

### Symbol of the Cube Root

Check the example below for detail:

5 Cube =

Thus, the cube root of 125 is 5. The number 125 is a perfect cube.

### Cube Roots (For Integer Results 1 Through 10)

The cube of a negative number will also be a negative number.

### Properties of Cube Roots

The table given below has the cubes of all the number between 11 to 20

A number having 1 in its unit digit, will also have 1 in the unit digit of their cubes.

1³  =  1

11³  =  1331

21³  =  9261

31³  =  29791

The cubes of 1, 4, 5, 6, 9, and 0 also have the same digits in its unit digits.

14³  =  2744

15³  =  3375

16³  =  4096

20³  =  8000

The cube of numbers ending 2 as unit digit will have 8 in its unit digit. Similarly, the cube of the numbers ending in unit digit 8 will have a unit digit 2.

12³  =  1728

18³  =  5832

The cube of the numbers with 3 as unit digit will have a unit digit 7. Similarly, the cube of numbers with the unit digit 7 will have a unit digit 3.

13³  =  2197

27³  =  19683

The cubes of all even numbers are even. Also, the cubes of odd numbers are all odd.

18³  =  5832 (even)

27³  =  19683 (odd)

The sum of the cubes of the first natural numbers m is equal to the square of their sum.

1³ + 2³ + 3³ + 4³  =  (1 + 2 + 3 + 4)²

1 + 8 + 27 + 64  =  (10)²

100  =  100

Cube root of any large number can be easily found in four ways:

Let’s know how to find the cube root of any number

Question 1)What is the cube root of 1728?

The factors of 1728 are given as,

Solution: Cube root of 27 is 3. (3 × 3 × 3)

Cube root of 125 is 5 (5 × 5 × 5)

Whenever you come across the word cube root the two words that come into our mind are cube and roots of a tree. Actually the concept is a bit similar in this sense, root actually refers to the primary source of origin. So we should think about what number you should refer to. That cube will give you a particular number that you are looking for.

Cube root of a number x is a number y only when  y×y×y= x. All the nonzero real numbers have one real cube root and along with it a pair of complex conjugate cube roots, and all non zero complex numbers have three different complex roots that are cube roots.

The cube root of any number in brief can be defined as the factor that we multiply by itself three times to get the particular numbers. Remember that the cube root of a number is exactly opposite of the cubing of a number.

The process of cubing is similar to squaring, only that the number is multiplied three times instead of two times as in squaring. The exponent used for cubes is 3, which is also denoted by the superscript³. Examples are 4³ = 4*4*4 = 64 or 8³ = 8*8*8 = 512 etc.

To find the volume of the cube, we have volume = side3, but if we want to find the side of a cube we have to take the cube root of the volume. Thus, we can say that the cube root is the inverse operation of cubing a number. The cube root symbol is 3√.

Let’s suppose we need to find the value of cube root of 2 is a value that is obtained by multiplying that number three times. It is expressed in the form of ‘3√2’. The meaning of cube root is basically the root of a number that is generated by taking the cube of another number. Hence, if the value of 3√2=x, then x =2 and we need to find the value of x.

We can define the cube root of a number as a special value that, when used in a multiplication exactly three times, gives us that number.For example, 3 × 3 × 3 equals 27, so the cube root of 27 is 3.

### The Cube Root Symbol

The special symbol given below signifies the «cube root», it is known to be the «radical» symbol (the symbol can be used for square roots) and with a little three to mean cube root.

You can use it like this, the cube root of 27 is : 3√27=3 (we say «the cube root of 27 equals to 3»)

### Find the Cube of Fraction

You can find the cube of a fraction the same way as you are finding the cube of a number. You just need to multiply the fraction three times. Just take the example of ⅔. You can get the cube of this fractional number by first multiplying the number in the numerator 3 times that is here you can first multiply the number 2 three times, 2×2×2 and as a result you will get 8 that is the Cube of the number 2 which is in your numerator.

Now move towards the denominator, here in your denominator there is 3, you can get its cube by multiplying it 3 times, that is 3×3×3 hereafter cubing the number 3 you are going to get 27 as a result, which is the Cube of your number 3, registered here as the denominator. So as a result you are going to get 8/27. This 8/27 is the Cube of your number ⅔ which you can get either by doing the multiplication of these numbers separately or you can do the multiplicity of the whole fraction three times, that is ⅔ × ⅔ × ⅔. Giving you the result of 8/27.

### You Can Also Cube Negative Numbers

Have a look at this:When we cube +5 we generally get +125:  +5 × +5 × +5 = +125

When we cube −5 we get the number −125:  −5 × −5 × −5 = −125. So the cube root of the number −125 is equal to −5

### What is the Meaning of Cube Root?

The cube root of a number a is that number which when multiplied by itself three times gives the number ‘a’ itself.

Let’s see for example,

23 =8, or the cube root of the number 8 is 2

33 = 27, or the cube root of the number 27 is 3

43 = 64, or the cube root of 64 is 4

53 = 125, or the cube root of 125 is 5

The symbol of the cube root is a

Thus, the cube root of 125 is represented as 3√125=5 and that of 27 can be represented as 3√27 equals 3 and so on.

We know that the cube of any number is found by multiplying that number three times. And the cube root of a number can be defined as the inverse operation of cubing a number.

If the cube of a number 63 = 216

Then the cube root of ∛216 is equal to 6.

Cube root of any largest number can be easily found in four ways:

### A Few Properties of Cube Root

A perfect cube of  an integer is that integer which is actually equal to some other integer raised to the third power. We refer to raising the number to the third power as cubing the number.

Perfect Cube :1 8 27 64 125 216 343 512 729 1000 .

### What is a Cube Root Calculator?

Cube root calculator is a tool that will help you to find the cube root of a particular number. This calculator is free. By using the cube root calculator you can find the cube root of a number free without giving any charge.

### How can you use a Cube Root Calculator?

What is the Cube Root of 30?

Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

Now we are getting closer, but slowly at this point, we can use a calculator and it says:

A perfect cube is the result of multiplying the same integer three times. Repeating the number 4 times three makes 64 and as a result 64 is a perfect cube.  64 has a cube root of 4. If a number can be decomposed into a product of the same three numbers, it is said to be a perfect cube.

A complete cube is the product of three identical numbers. To see if the number (for example n) is a suitable cube,multiply it by three times and check if the resulting number is the same as ‘n’ if yes then it is a perfect cube

One, eight, twenty-seven, and sixty-four are examples of perfect cubes. A perfect square is one that can be made by multiplying two numbers together. Both positive and negative numbers can be used to make perfect cubes. Because it is the product of multiplying -4 three times, -64 is a perfect cube.

When we say a number has been cubed, we mean that it has been multiplied three times. The process of cubing a number  is reversed by its root. When the number 5 is cubed, for example, we obtain 55, which is 125. The cube  root of 5 is 125. This is because multiplying the number 5 three times gives the result of 125.It is the same as the square root symbol with an addition of ‘3’ to indicate it is a cube root .The cube root of a number can be stated in exponent form as (number

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### Cube root table

Cube root table is a table consisting of a list of numbers and their cube roots. Before we write the cube root table, it is very important for all of us to understand what is the perfect cube root chart and cube of a number. The cube of any real number can be defined as that number which is obtained by multiplying a number by itself twice or by raising its power to 3. At the same time, the cube root of any number is that number which when raised to the power 3 gives the answer as the number whose cube root is to be determined.

### Perfect Cube Root Chart

Cube root of a number can also be exponentially represented as the number raised to the power ⅓. If ‘x’ is any real number, then its cube root is represented as (x) or ∛x. Perfect cubes are the numbers that are obtained when natural numbers are multiplied by itself twice. All the perfect cube numbers have a square root equal to a natural number. The perfect cube root chart of the first 25 perfect cube numbers is represented in the table below.

While it is very important to know the cube roots of the numbers, it is also very important to know the table of cubes from 1 to 50 (i.e. first 50 natural numbers) in order to make our Mathematical computations easy and efficient without a calculator. The table of cubes from 1 to 50 is given below for the reference of students and facilitators.

### Table of Cubes from 1 to 50

Any student or a facilitator who is keen about Mathematics will find his or her own ways to memorize the cube root list 1 to 100. Though it may seem a little difficult at the initial days, the task of remembering the cube root list 1 to 100 becomes more and more easy through continuous practice and periodic revision.

### Cube Root List 1 to 100

Step 1: Starting with the smallest prime number, prime factorise the given number (2).

Step 2: Once the prime factorization is complete, group together all three identical factors.

Step 3: Repeat Step 3 for all sets of the same three factors in the group. The provided number is not a perfect cube if any factors are left over that do not fit into a group of three identical factors. The provided integer is otherwise a perfect cube.

### Fun Fact About Cube Root

There is a method for determining whether or not greater integers are perfect cubes. To check, find the sum of the number’s digits repeatedly and see if it’s either 0, 1, 8, or 9. If it’s any of these, it could be a perfect cube, although that isn’t always the case.

### Perfect Cube Formula

The perfect cube formula is used to determine whether or not a number is a perfect cube. Let’s say we have a number, x, that equals yy. Every composite number can be written as the product of the powers of its prime elements, according to the fundamental theorem of arithmetic. The number is considered to be a perfect cube if the power of all prime factors is a multiple of three.

From 1 to 50, this is a list of perfect cubes.

In the table below, the perfect cubes of integers 1 to 50 are presented. Every integer is multiplied three times by itself to get the ideal cubes.

## Definition – What is a cubed root?

The cubed root of a number is the number that, when multiplied by itself three times (number x number x number), gives the original number.

For example, the cubed root of 27 is 3, as 3 x 3 x 3 = 27.

The cubed root of 125 is 5, as 5 x 5 x 5 = 125.

Unlike the square root, the cubed root is always positive.

The opposite of the cubed root is a cubed (power of 3) calculation.

In geometry cubed root can be used to find the length of a side of a cubed when the volume is known.

## Formula – How to calculate the cubed root of a number

There is no quick formula to calculate a cubed root. Most calculators use some form of trial and error.

Most calculators use trial and error to find a cubed root. Trial and error works well for perfect cubes. It can be very time consuming for non-perfect cubes because there are many decimal places.

To find cubed root by trial and error:

Example – Find the cubed root of 512 using trial and error:

### Method 2 – Quickly find roots from perfect cube numbers

This method makes it quicker to find the root of a perfect cube number. If the number is not a perfect cube root number, however, this method won’t give an answer.

## How do you type cubed root?

The word root means the source, so by the literal definition, cube root means the cube of any number must be taken to get the original number. The product of a number multiplied thrice (three times) by itself is the  of that specific number. A cube root is the inverse of the cube of any number, so if we take the cube root of a number, then the answer will always be the original number.

Figure 1 – Cube Root of 1

Let us take a number 2 to understand the concepts of cube roots. We know that when this number is multiplied three times, we get 8 as 2 × 2 × 2 = 8. So the number 8 is the cube of 2, which can be written as:

23 = 8

Now to take the cube root of this number, we will use the inverse method. As discussed above cube root is a factor of the number itself:

The cube root of 8 is, therefore, 2.

To solve the mathematical questions, we use the cube root formula to ease us. We can do this by finding the prime factorization of the number. After that, we take the cube root to the answer. Let us take the example of 64:

64 = 4 × 4 × 4

Now applying the cube root formula:

Thus, 4 is the cube root of 64.

## When we multiply 3 same integers with each other then the resulting cube is known as the perfect cube and similarly, the square root will be a whole number integer. Such numbers are known as cube numbers.

Figure 2 – Perfect Cube of A

Let us take the example of the number 27:

33 = 3 × 3 × 3 = 27

Now taking the cube root:

Hence 27 is a perfect cube.

Not every number is a perfect cube if we take the example of the number 34 then we get:

This is because it does not satisfy the condition in which the cube root must be a whole number integer.

Let us consider a number x which is a negative number so to take its cube root we will rewrite it again while separating the negative sign from it:

After separating the negative sign we do prime factorization on x and list all possible factors. We will then make groups of three and multiply them together so that we can get the cube root of x.

Note that the cube root of a negative number will always be a negative number and similarly the cube root of a positive number is also positive.

## Examples of Cube Root

Hailey is a part of a gaming club where you have to collect badges to level up. The highest record in this club is 216 badges. Find the number of Hailey’s badge if her level is the cube root of the highest record holder. Then tell how many more badges she needs to earn to become the record holder.

From the statement given above, we can say that the highest record is 216 so:

So this tells that Hailey is on level 6 at the moment and to beat the record she needs:

216 – 6 = 210

210 badges are required for Hailey to reach the record but to break it she needs 1 more badge:

210 + 1 = 211 badges

Todd wants to find the cube root of 2,744 for his math assignment. Help him find it.

We will solve this by using the prime factorization method so:

Figure 3 – Factor Tree Representing Prime Factorization of 2744

2744 = 2 × 2 × 2 × 7 × 7 × 7

Now we will group them so that we have 3 numbers in each pair:

2744 = (2 × 2 × 2) × (7 × 7 × 7)

2744 = 23 × 73

= 2 × 7

So the Cube root of 2744 is the number 14.

First, we use the prime factorization method:

512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Now we will group them together:

512 = (2 ×2 × 2) × (2 × 2 × 2) × (2 × 2 × 2)

512 = 23 × 23 × 23

Hence 512 is a Perfect Cube.

48 = 2 × 2 × 2 × 2 × 3

48 = (2 × 2 × 2) × (2) × (3)

48 = 23 × 2 × 3

Hence 48 is not a Perfect Cube.

Images/mathematical drawings are created with GeoGebra.

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